ALGEBRA QUALIFYING EXAM
September 9, 2000
Do 7 of the 9 problems.
1. Prove that there is no simple group of order 36.
2. Prove that every nontrivial group has a factor group which is simple.
3. Exhibit an irreducible polynomial in Q[x] which has 3
√2 + 2
√3 as a root.
4. Prove that if Ris a commutative ring with unity, then Ris Noetherian
if and only if the polynomial ring R[x] is Noetherian.
5. Prove that the set of algebraic integers (roots of monic polynomials in
Q[x]) form a subring of the algebraic numbers.
6. Let pbe a prime and let G=GL2(Z/p), the group of invertible 2 ×2
matrices with entries in Z/p.
(a) find a Sylow p-subgroup of G.
(b) Find the number of Sylow p-subgroups of G.
7. State and prove the Chinese Remainder Theorem.
8. Prove that x5−6x+ 3 has exactly 3 real roots and then use this fact to
calculate its Galois group over Q.
9. Let Abe an n×nmatrix with entries in C. Show that if A2=A, then
there are subspaces Kand Lof Cnso that L⊕K=Cn, and so that A|L
is the identity and A|K≡0.
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